Analyticity of the correlation length in subcritical FK-percolation and Potts models

Source abstract

We prove that the correlation length of FK-percolation and Potts models on $\mathbb{Z}^d$ is analytic in the whole subcritical regime of parameters. Our proof makes use of the identification of the polar dual of the mass with the solution of an equation involving the leading eigenvalue of the FK--Ruelle operator, introduced in the context of Ornstein--Zernike theory in Campanino, Ioffe and Velenik (Probab. Theory Relat. Fields, 2003), and crucially relies on Kato's theory of analytic perturbation of linear operators. As a consequence of this result, we prove that the correlation length cannot be invariant by rotation except at countably many values of the temperature parameter in the subcritical phase.